Fourier coefficients of a Hecke-ring eigenvector at the good primes #
Let F ∈ M_k(N, χ) (or S_k(N, χ)) be an eigenvector of the Γ₀(N) Hecke-ring generator at a
prime p ∤ N, acting through heckeRingHomCharSpace (heckeRingHomCuspCharSpace), with
eigenvalue c. Through the identification of that generator with the classical T_p
(heckeRingHomCharSpace_heckeTGeneratorGamma0, heckeRingHomCuspCharSpace_heckeTGeneratorGamma0)
and the coefficient formula a_m(T_p F) = a_{pm}(F) + χ(p) p^{k−1} a_{m/p}(F) of
HeckeSlash/Recurrence.lean, the eigenvector equation becomes a recurrence on the Fourier
coefficients of F alone:
a_{pm}(F) = c · a_m(F) − χ(p) p^{k−1} a_{m/p}(F), the last term present only when p ∣ m.
Running it along the least prime factor shows that a form which is an eigenvector at every
prime outside an auxiliary level L (a multiple of N; for L ≠ 0 these are all but finitely
many primes) and has a₁ = 0 has a_n = 0 at every nonzero index n coprime to L — at
n = 0 as well when it is a cusp form. This is the form
in which strong multiplicity one consumes eigen-ness (Miyake's Theorem 4.6.12 assumes agreement
at the indices prime to such an L): the difference of two newforms whose eigenvalues agree
outside L has a₁ = 1 − 1 = 0, so its coefficients at the indices prime to L all vanish,
and the descent argument then places it in the old subspace.
Main results #
heckeTNat_eq_smul_iff_forall_qExpansion_coeff_prime_mul: onM_k(N, χ)the recurrence characterises the eigen-relationTₚ F = c • F— the converse holds too, because a modular form is determined by itsq-expansion. This is the direction Diamond–Shurman's Proposition 5.8.5 needs.heckeTCuspNat_eq_smul_iff_forall_qExpansion_coeff_prime_mul: its cusp-form case, the spellingIsEigenformAwayFromLeveluses.qExpansion_coeff_prime_mul_of_heckeRingHomCharSpace_heckeTCompositeGamma0_eq_smuland its cusp-form counterpart…_of_heckeRingHomCuspCharSpace_…: the coefficient recurrence of an eigenvector, onM_k(N, χ)and onS_k(N, χ); the cusp-form one is the forward half of the characterisation above.qExpansion_coeff_eq_zero_of_forall_prime_heckeRingHom_of_one_eq_zero_of_ne_zero_of_coprimeand…_heckeRingHomCuspCharSpace_…: a form eigen at every prime away from a multipleLofN, witha₁ = 0, hasa_n = 0at everyncoprime toL(andn ≠ 0in the modular-form case).heckeTNat_eq_smul_of_heckeRingHomCharSpace_heckeTCompositeGamma0_eq_smuland its cusp-form counterpart: a Hecke-ring eigenvector at a good prime is an eigenvector of the classical operatorheckeTNat(resp.heckeTCuspNat).heckeRingHomCuspCharSpace_heckeTCompositeGamma0_eq_smul_of_heckeTCuspNat_eq_smul: the converse onS_k(N, χ), at every prime, whether or not it divides the level.
References #
- F. Diamond and J. Shurman, A first course in modular forms, Proposition 5.3.1 and §5.8 — in particular Proposition 5.8.5, whose engine at a single prime is the characterisation stated here.
- T. Miyake, Modular forms, §4.6.
A ring eigenvector at a good prime is an eigenvector of the classical T_p, on
M_k(N, χ): at a prime the Hecke ring's action and heckeTNat are the same operator, so the two
eigen-equations are the same statement.
A ring eigenvector at a good prime is an eigenvector of the classical T_p, on
S_k(N, χ).
An eigenvector of the classical T_p is a ring eigenvector, on S_k(N, χ), at every
prime p, whether or not p ∣ N (for p ∣ N read T_p = U_p): the converse of
heckeTCuspNat_eq_smul_of_heckeRingHomCuspCharSpace_heckeTCompositeGamma0_eq_smul, the same
equation read back through the coercion.
The recurrence characterises the eigen-relation #
The coefficient recurrence characterises the eigen-relation Tₚ F = c • F at a good
prime, on M_k(N, χ). For F ∈ M_k(N, χ) and p ∤ N, the relation Tₚ F = c • F holds
exactly when the Fourier coefficients of F satisfy
a_{pm}(F) = c · a_m(F) − χ(p) p^{k−1} a_{m/p}(F) at every m, the last term present only when
p ∣ m.
This is a statement about the equation, not about eigenvectors: F = 0 satisfies both sides for
every c, and a consumer wanting a genuine eigenvector supplies F ≠ 0 itself.
The coefficient recurrence characterises the eigen-relation Tₚ F = c • F at a good
prime, on S_k(N, χ): the cusp-form case of
heckeTNat_eq_smul_iff_forall_qExpansion_coeff_prime_mul, in the spelling
IsEigenformAwayFromLevel uses.
The coefficient recurrence of an eigenvector at a good prime, on M_k(N, χ). If the ring
generator at p ∤ N acts on F ∈ M_k(N, χ) by the scalar c, then
a_{pm}(F) = c · a_m(F) − χ(p) p^{k−1} a_{m/p}(F), the last term present only when p ∣ m.
The coefficient recurrence of an eigenvector at a good prime, on S_k(N, χ). If the ring
generator at p ∤ N acts on F ∈ S_k(N, χ) by the scalar c, then
a_{pm}(F) = c · a_m(F) − χ(p) p^{k−1} a_{m/p}(F), the last term present only when p ∣ m.
Coefficient vanishing from the prime eigenvalues, on M_k(N, χ). Let L be a multiple of
N. A form F ∈ M_k(N, χ) that is an eigenvector of the ring generator at every prime p ∤ L
and has a₁(F) = 0 has a_n(F) = 0 at every n ≠ 0 coprime to L.
For L ≠ 0, the auxiliary level is the finite slack of strong multiplicity one: eigen-ness is
assumed only away from finitely many primes beyond those dividing N.
Coefficient vanishing from the prime eigenvalues, on S_k(N, χ). Let L be a multiple of
N. A cusp form F ∈ S_k(N, χ) that is an eigenvector of the ring generator at every prime
p ∤ L and has a₁(F) = 0 has a_n(F) = 0 at every n coprime to L — with no n ≠ 0
hypothesis, unlike the modular-form version, since a cusp form already has a₀ = 0. This is the
form strong multiplicity one consumes.